If a : b = 2:3, b: c= 4:5 and c:d=6:7, then a:d=

Options

A. `12 :35`
B. `24 :35`
C. `16 :35`
D. `24 :25`

Answer

To solve the problem, we need to find the ratio \( a:d \) given the ratios \( a:b = 2:3 \), \( b:c = 4:5 \), and \( c:d = 6:7 \).

### Step-by-Step Solution:

1. **Write down the given ratios**:
- \( a:b = 2:3 \)
- \( b:c = 4:5 \)
- \( c:d = 6:7 \)

2. **Express each variable in terms of a common variable**:
- Let \( a = 2x \) and \( b = 3x \) from the first ratio.
- From the second ratio \( b:c = 4:5 \), we can express \( b \) in terms of \( c \):
\[ b = 4y \quad \text{and} \quad c = 5y \]
- Since \( b \) is the same in both expressions, we can equate them:
\[ 3x = 4y \implies y = \frac{3x}{4} \]
- Substitute \( y \) back to find \( c \):
\[ c = 5y = 5 \left(\frac{3x}{4}\right) = \frac{15x}{4} \]

3. **Express \( c \) in terms of \( d \)**:
- From the third ratio \( c:d = 6:7 \):
\[ c = 6z \quad \text{and} \quad d = 7z \]
- Equate \( c \) from the previous step:
\[ \frac{15x}{4} = 6z \implies z = \frac{15x}{24} = \frac{5x}{8} \]
- Substitute \( z \) back to find \( d \):
\[ d = 7z = 7 \left(\frac{5x}{8}\right) = \frac{35x}{8} \]

4. **Now we have expressions for \( a \) and \( d \)**:
- \( a = 2x \)
- \( d = \frac{35x}{8} \)

5. **Find the ratio \( a:d \)**:
\[ a:d = \frac{2x}{\frac{35x}{8}} = \frac{2x \cdot 8}{35x} = \frac{16}{35} \]

6. **Final Result**:
- Therefore, the ratio \( a:d = 16:35 \).